On a bounded domain, let $u \in L^2\left(0,T;H^1\right) \cap H^1\left(0,T;H^{-1}\right)$ be weak solution to the heat equation $$u_t - \Delta u = 0$$ with the BC $\partial_\nu u = 0$ and some initial data, i.e. $$\langle u_t, v \rangle + \int_\Omega \nabla u \nabla v = 0$$ holds for all $v \in L^2(0,T;H^1)$ for a.a. $t$.
Suppose we know that $u \geq 0$ a.e.
I want to test the equation with $\log(u)$. I don't think this is possible immediately since I can't show that $\nabla \log(u) = \frac{1}{u}\nabla u$ is in $L^2$. Is there some way to approximate this test function?